In the previous visit we took to the mysterious room we went around the side and up the stair to the top, where we took a look at one Japanese approach to the sine function of a right angled triangle.
Today we will head towards the other door to the mysterious room, the door that the carpenter might normally take. Again, we pass by a sign on the way:
Setting aside what the sign says for the moment, notice the interesting form this object takes: besides the lovely carving and lacquer work, it is in the form of a splayed box. It would appear that the box has a regular plan with 90˚ turns at the corners, and that the sides of the box slope an equal amount on all sides. Further, the sides connect to each other with butt joints, not miters.
Further along our walk, we come to a small stand, and again notice a small splayed box on top:
These sort of splayed boxes are called by various names - battered boxes, hoppers, funnels, and so forth. In Japanese this form often goes by the name of asa-gao-gata, 朝顔形, or 'morning glory shape' boxes. In case the reader hasn't seen a Morning Glory flower before, here's a look at one variety:
Interesting that the Morning Glory has pentagonal symmetry, like a lot of flowers. The asa-gao-gata box can be any number of sides you like, however the most common and simplest to deal with has four sides, each side sloped the same amount. That's the kind of box I will be discussing here.
If you think about it, the hopper form is the same as that of a hipped roof, only upside down. In fact, if you made a lid for a box in the form of an inverted hopper, you may as well say you are making a lid in the form of a hipped roof shell. Thus, the geometry that applies to the angles in which the boards of the hopper connect at their corners will also have applications for parts in a hip roof that are oriented in a similar manner. Most obviously, boards which are employed to plank over the rafters, would share the geometry of the hopper boards where they meet atop the hip rafter, but there are many other possibilities, both in Western and Eastern carpentry.
A hopper is the foundation stone of studying compound joinery, and these are forms with a wide variety of applications. Here's a hopper I made for use as a water stone soaking pond and sharpening station:
Time is running short today, so I'll have to put off considering the geometry of the hopper box in further detail and how we might determine the cut angles.
There are two conditions only: butt joints and miter joints. Further, in sum total there are only three angles we need to figure out for most constructions:
1. The face cut angle 2. The edge cut angle for butt joint 3. The edge cut angle for miter joint
If the hopper is to have through-tenoned connections at the joins, there is an additional 4th special angle. Typically of course, the hopper would be joined with the same form of joint at all four corners, if only to be nailed together, and thus we would only need to figure out two angles (1 and 2, or 1 and 3) to make the piece.
In case it isn't clear: if the boards did not slope, there would be no compound joinery problem to deal with -- the face cut angle would be 90˚ and the edge cut angle would be either 45˚ (miter) or 90˚ (butt joint).
All the angles can be figured with simple drawings which I show in the essay The Art of Japanese Carpentry Drawing, Volume II(available for purchase via the link at the top right of the page). It helps to be methodical I find when entering into a topic - compound joinery - that can get somewhat complicated after a while, and thus I have written a very complete description for how to lay out and build hoppers.